抄録
This is a survey paper on various methods developed in recent years for obtaining asymptotic expansions for the nonnull distributions of multivariate test statistics. In Part I, two methods are discussed: (1) one by the Taylor expansion, including studentization; (2) one by the expansion of the exact characteristic functions using the formulae for weighted sums of zonal polynomials. Part II will be given in the separate paper in the next issue, and contain (3) the method based on the system of partial differential equations satisfied by the hypergeometric functions, (4) the method by an expansion of test statistics in terms of normal variates, and (5) the method based on the representation of an orthogonal matrix by a skew symmetric matrix. In the discussion, emphasis is on the basic idea of the deriving techniques, and details of computation and final forms of individual asymptotic expansions explained as examples to clarify the idea are omitted; appropriate references, however, are given.
